From the Bandaranaike International Airport, you can get to Colombo by an Uber-like Sri Lankan service Pickme, regular taxi, train, or bus.
Pickme is the most convenient option as you can book your car in the Pickme app, choose the type of car you want, and the cost from the airport to the city center is only around 1,600 LKR (9 USD). It's much cheaper than regular taxis.
The regular taxi drive takes around 30 minutes and costs at least 2,500 LKR (14 USD). You can either find the official taxi stand when you exit the luggage claim, or take the offer of one of the random drivers who'll approach you after you exit the luggage claim.
The train station is not far from the airport, only about a 100 LKR (0.50 USD) tuk-tuk ride. Trains are quite cheap, as trains are often delayed and crowded, this isn’t a very attractive and secure option.
Buses are another cheap option. Bus E3-187 is the big air conditioned bus that runs between the airport and Colombo Fort Station, which is the main train and bus station in Colombo. Cost is 130 LKR (0.70 USD) one way to the city, you pay by cash on the bus itself. There's no fixed schedule, the bus runs once every 30 minutes or so. The trip to Colombo takes about 30min-1 hour depending on traffic. The last bus leaves at around 8 pm from the airport. After 8 pm, there will still be other buses, but they may not be from the same boarding area.
You can check the schdule for buses and trains here.